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A Note on Coloring of 3/3-Power of Subquartic Graphs

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A Note on Coloring of 3/3-Power of Subquartic Graphs AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 79(3) (2021), Pages 454{460 3 A note on coloring of 3-power of subquartic graphs Mahsa Mozafari-Nia Moharram N. Iradmusa Department of Mathematical Sciences Shahid Beheshti University G.C. P.O. Box 19839-63113, Tehran Iran Abstract 1 For any k 2 N, the k-subdivision of a graph G is a simple graph G k , which is constructed by replacing each edge of G with a path of length k. In [M.N. Iradmusa, Discrete Math. 310 (10-11) (2010), 1551{1556], the mth power of the n-subdivision of G was introduced as a fractional m power of G, denoted by G n . F. Wang and X. Liu in [Discrete Math. 3 Algorithms Appl. 10 (3), (2018), 1850041] showed that χ(G 3 ) ≤ 7 for 3 any subcubic graph G. In this note, we prove that the 3 -power of every subquartic graph admits a proper coloring with at most nine colors. We 3 conjecture that χ(G 3 ) ≤ 2∆(G) + 1 for any graph G with maximum degree ∆(G) ≥ 2. 1 Introduction All graphs we consider in this note are simple, finite and undirected. We men- tion some of the definitions that are referred to throughout this note, and for other necessary definitions and notation we refer the reader to a standard text-book [2]. Minimum degree, maximum degree and maximum size of cliques of a graph G are denoted by δ(G), ∆(G) and !(G), respectively. For each vertex v 2 V (G), the neigh- bors of v are the vertices adjacent to v in G. The neighborhood NG(v) of v is the set of all neighbors of v in G. Any vertex of degree k is called a k-vertex and any path of length k is called a k-path. Also, a path P : v1; : : : ; vk is a simple path if the degrees of all vertices v2; : : : ; vk−1 are 2, and a cycle C is called a loop cycle if all of its vertices are 2-vertices except one that is a 3-vertex. Let G be a graph and k be a positive integer. The k-power of G, denoted by Gk, is defined on the vertex set V (G) by adding edges joining any two distinct vertices x and y with distance at most k. Also the k-subdivision of G, denoted by 1 G k , is constructed by replacing each edge xy of G with a path of length k, say Pxy. These k-paths are called superedges. We denote a vertex by (xy)l if it belongs to Pxy and has distance l from the vertex x, where l 2 f0; 1; 2; : : : ; kg. Note that ISSN: 2202-3518 c The author(s) M. MOZAFARI-NIA ET AL. / AUSTRALAS. J. COMBIN. 79 (3) (2021), 454{460 455 1 (xy)l = (yx)k−l, x = (xy)0 = (yx)k and y = (yx)0 = (xy)k. Any vertex (xy)0 of G k is called a terminal vertex (or briefly t-vertex) and any of the remaining vertices is called an internal vertex (or briefly i-vertex). The fractional power of graphs was first introduced in [5] as follows. m Definition 1.1 Let G be a graph and m; n 2 N. The graph G n is defined to be the m 1 m m-power of the n-subdivision of G. In other words G n = (G n ) . m m We denote the set of terminal vertices of G n by Vt(G n ) and the set of internal m 1 2 vertices by Vi(G n ). It is worth noting that G 1 = G and G 2 = T (G), where T (G), the total graph of G, is the the graph whose vertex set is V (G) [ E(G), in which two vertices are adjacent if and only if they are adjacent or incident in G [1]. As usual, a proper k-coloring of G is a mapping from V (G) to f1; : : : ; kg, where any two adjacent vertices have distinct colors. The chromatic number of a graph G is the minimum integer k for which G has a proper k-coloring, and is denoted 00 2 by χ(G). By the definition of a total graph, χ (G) := χ(T (G)) = χ(G 2 ). In 1965, Behzad [1] conjectured that χ00(G) never exceeds ∆(G) + 2. By virtue of Definition 2 1.1, one can show that !(G 2 ) = ∆(G) + 1 and the Total Coloring Conjecture can be reformulated as follows. 2 2 Conjecture 1.2 For any simple graph G, χ(G 2 ) ≤ !(G 2 ) + 1. There is a relation between an incidence coloring of a graph G and a vertex 3 coloring of G 3 , which is one of the motivations of this note. The concept of incidence coloring was introduced by Brualdi and Massey in 1993. Definition 1.3 [3] Let G = (V; E) be a multigraph. An incidence of G is a pair (v; e) where v 2 V (G), e 2 E(G) and e is incident with v. Let I be the set of incidences of G. An incidence graph of G, denoted by I(G), has its vertex set V (I(G)) = I such that two incidences (v; e) and (w; f) are adjacent in I(G) if one of the following holds: (1) v = w, (2) e = f, (3) the end-vertices of e or f are v and w. Definition 1.4 [3] An incidence coloring σ of G is a map from I to the color set C such that adjacent incidence pairs are assigned different colors. If σ : I −! C is an incidence coloring with jCj = k, then we say that σ is a k-incidence coloring of G. The incidence chromatic number of G, denoted χi(G), is the smallest k for which there exists a k-incidence coloring of G. In [3], it is proved that for a graph G with maximum degree ∆, χi(G) ≤ 2∆. Also, by definition of the fractional power of a graph, the incidence graph I(G) is M. MOZAFARI-NIA ET AL. / AUSTRALAS. J. COMBIN. 79 (3) (2021), 454{460 456 3 the subgraph of G 3 induced by the set of internal vertices. So we have χi(G) = 3 3 3 3 3 χ(G 3 [Vi(G 3 )]) ≤ χ(G 3 ). In addition, the partition fVt(G 3 );Vi(G 3 )g of the vertices 3 of G 3 implies that 3 3 3 3 3 χ(G 3 ) ≤ χ(G 3 [Vt(G 3 )]) + χ(G 3 [Vi(G 3 )]) = χ(G) + χi(G): 3 Also, in [6] it was proved that if ∆(G) ≥ 3, then χ(G 3 ) ≤ χ(G) + χi(G) − 1. 3 In this note, we are investigating the chromatic number of G 3 . When ∆(G) = 1, 3 one can easily show that χ(G 3 ) = 4, and by applying the following theorem, which 3 was proved in [5], we can prove that χ(G 3 ) ≤ 5 for any graph G with ∆(G) = 2. Theorem 1.5 If m; n; k 2 N and k ≥ 3, then ( nk m nk m ≥ n 2 (i) χ(Ck ) = nk nk d nk e m < 2 ; b m+1 c m n (ii) χ(Pk ) = minfm + 1; (k − 1)n + 1g. 3 In [8], Wang and Liu proved that χ(G 3 ) ≤ 7 for any subcubic graph G. Recall that a graph G is subcubic if ∆(G) ≤ 3. Recently, a simple proof of this result was given in [6] by using the following theorem about the 5-colorability of the incidences of any subcubic graph. Theorem 1.6 [7] For any subcubic graph G, we have χi(G) ≤ 5. The graph G is called subquartic if ∆(G) ≤ 4. Also any 4-regular graph is known as a quartic graph. The main theorem of this note is stated as follows. 3 Theorem 1.7 Let G be a subquartic graph. Then χ(G 3 ) ≤ 9. Remark 1.8 In [3], it was conjectured that χi(G) ≤ ∆(G) + 2 for every graph G. This was disproved by Guiduli in [4] who showed that Paley graphs with sufficiently large maximum degree have incidence chromatic number at least ∆+Ω(log ∆). How- ever, this conjecture seems to hold for graphs with small maximum degree. Theorem 1.6 shows that the conjecture is true for cubic graphs. It remains an open problem whether the conjecture is true for quartic graphs. But if the conjecture holds for all quartic graphs, then easily we can prove Theorem 1.7 by use of the inequality 3 χ(G 3 ) ≤ χ(G) + χi(G) − 1. So Theorem 1.7 provides evidence that the conjecture may be true for all quartic graphs. Considering the results for cycles, paths, subcubic and subquartic graphs, we 3 conjecture that 2∆(G) + 1 colors suffice for the proper coloring of G 3 when G is a graph with maximum degree at least two. 3 Conjecture 1.9 Let G be a graph with ∆(G) ≥ 2. Then χ(G 3 ) ≤ 2∆(G) + 1. M. MOZAFARI-NIA ET AL. / AUSTRALAS. J. COMBIN. 79 (3) (2021), 454{460 457 The clique number of the fractional power of a graph was obtained in [5] for powers 3 less than one. As mentioned in [8], one can easily show that !(G 3 ) = ∆(G)+2 when 3 ∆(G) ≥ 2, and !(G 3 ) = 4 when ∆(G) = 1. Therefore, if Conjecture 1.2 holds, we 00 2 3 conclude that χ (G) = χ(G 2 ) ≤ χ(G 3 ).
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